Measurements of photoluminescence and its dependence on hydrostatic pressure are performed on a set of InN/nGaN superlattices with one InN monolayer, and with different numbers of GaN monolayers (n from 1 to 40). The emission energies, EPL, measured at ambient pressure, are close to the value of the band gap, Eg, in bulk GaN, in agreement with other experimental findings. The pressure dependence of the emission energies, dEPL/dp, however, resembles that of the InN energy gap. Further, the magnitudes of both EPL and dEPL/dp are significantly higher than those obtained from abinitio calculations for 1InN/nGaN superlattices. Some causes of these discrepancies are suggested...Detailed analysis of the electronic band structure of 1InN/5GaN superlattice is performed showing that the built-in electric field plays an important role in the mInN/nGaN structures. It strongly influences the valence- and conduction-band profiles and thus determines the effective band gap.